Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/107

 ON ELECTRIC DENSITY.

Distribution in Three Dimensions.

64.] Definition. The electric volume-density at a given point in space is the limiting ratio of the quantity of electricity within a sphere whose centre is the given point to the volume of the sphere, when its radius is diminished without limit.

We shall denote this ratio by the symbol $$\rho$$, which may be positive or negative.

Distribution on a Surface.

It is a result alike of theory and of experiment, that, in certain cases, the electrification of a body is entirely on the surface. The density at a point on the surface, if defined according to the method given above, would be infinite. We therefore adopt a different method for the measurement of surface-density.

Definition. The electric density at a given point on a surface is the limiting ratio of the quantity of electricity within a sphere whose centre is the given point to the area of the surface contained within the sphere, when its radius is diminished without limit.

We shall denote the surface-density by the symbol $$\sigma$$.

Those writers who supposed electricity to be a material fluid or a collection of particles, were obliged in this case to suppose the electricity distributed on the surface in the form of a stratum of a certain thickness $$\theta$$, its density being $$\rho_0$$, or that value of $$\rho$$ which would result from the particles having the closest contact of which they are capable. It is manifest that on this theory

When $$\sigma$$ is negative, according to this theory, a certain stratum of thickness $$\theta$$ is left entirely devoid of positive electricity, and filled entirely with negative electricity.

There is, however, no experimental evidence either of the electric stratum having any thickness, or of electricity being a fluid or a collection of particles. We therefore prefer to do without the symbol for the thickness of the stratum, and to use a special symbol for surface-density.

Distribution along a Line. It is sometimes convenient to suppose electricity distributed on a line, that is, a long narrow body of which we neglect the